But, what if is non-separating (but still 2-sided)? Then, there are two natural maps representing , where . Associated to , we have a map , , which maps a curve to its signed (algebraic) intersection number with .

Let be a covering map corresponding to . Then,

This has a shift-automorphism . We can now recover :

Defintion. If are injective homomorphisms, then let

Let be the shift automorphism on . Now, is called the HNN (Higman, Neumann, Neumann) Extension of over . We often realize as , where and . It is easy to write down a presentation:. is called a stable letter.

HNN extensions are pushouts in the category of groupoids. This was first pointed out to me by Ronnie Brown on math.SE. (The pushout diagram is a little annoying to and I’m not sure if anyone’s reading this. Exercise?)